Trigg Numbers

A Numerical Treasure Hunt

     Many years ago an important math scholar made a little study of a special group of numbers. Mr. Trigg, the mathematician, computed all the numbers less than 10,000 that could be formed by adding a square number to a cubic number. This simply means, all numbers that come from this algebraic expression:

N2 + M3

when you replace N and M with some appropriate whole numbers.

     He, of course, found many numbers in this way. Afterall there is no end to the list of whole numbers; hence, there is no end to the lists of squares and cubes of those numbers. That's why Mr. Trigg decided to limit himself to an analysis of only those that gave sums smaller than 10,000. But even at that, he found a lot of interesting results.

     For example, he found many numbers that were palindromes, tautonyms [numbers of the form abab], or those formed in two different ways, and yes, even some that came out to be squares themselves. Here are some examples of what we're talking about:

Palindrome: 4994 = 92 + 173

Tautonym: 5252 = 342 + 163

Two ways: 1025 = 322 + 13 = 52 + 103

Square: ` 1225 = 152 + 103 [= 352]

     Now that you have the general idea, here's where the fun begins!

Can you find some more "jewels" in Mr. Trigg's
treasure box of numbers?

     Of course, you have to decide what you are going to look for. We've just mentioned four types. So that's a start at least. But there are other "types" of numbers, like primes. Can you find some Trigg Numbers -- that's what we here in the World of Trotter Math have chosen to call numbers that are formed in this special way -- that are primes? This means they should be called "Trigg Primes", right? A small example of a Trigg prime would be

127 = 102 + 33

     There are other special formulations that we like here at WTM, one of which goes this way: Look for a 4-place number such that when you "split it in half", you get consecutive 2-place numbers. Got that? Here's an example:

4344 = 402 + 143

     And you'll surely agree that 43 and 44 are consecutive numbers.

     One particular favorite of ours is this bright little gem:

3456 = 92 + 153

     We've found one more number like that beauty (in which the digits are "in order"); we leave it as "homework" for you. And there are some other numbers in which the consecutive digits are there, just a little disordered!

1243 = 302 + 73

1432 = 332 + 73

     Another category we like is called Zerger Numbers, in honor of a friend of ours, Monte Zerger. Due to a discussion with him, we've decided his numbers take the shape of aabb. And can you believe it? There exist some "Zerger-Trigg Numbers", or should that be "Trigg-Zerger Numbers"? We don't know yet. Anyway, here is a nice example:

1144 = 122 + 103

     See? And there are more out there, just waiting for you to dig them up.

     And so it goes... Maybe you can invent your own special type; that's what's really fun.

     By the way, here's a special little challenge for you to chew on. It should be clear that the smallest number Trigg included in his study was 2: 12 + 13. But what was the largest? (Remember: it would be less than 10,000.) And what are the values for N and M in this case?

How to Hunt

     To find more Trigg Numbers is easy; just choose any values you wish for N and M, substitute them in the expression and calculate the results. The trick is to find numbers that are really special and interesting. You could take out paper and pencil and start choosing at random, but that would certainly take a long time, be boring, and probably you would make a lot of errors. (I know it would happen with me!)

     However, a calculator should speed things up considerably, don't you think? And if you do things in order and keep a record as you go along, progress will be easier. One nice way to do this kind of hunting uses a special property of the odd numbers. Notice this pattern:

		 1 = 1
		 4 = 1 + 3
		 9 = 1 + 3 + 5
		16 = 1 + 3 + 5 + 7
		25 = 1 + 3 + 5 + 7 + 9

		etc.

If we find the sum of the first n odd numbers, we get n2. So if we choose some particular cube, say 27, then add 1, add 3, add 5, etc., even with a very simple calculator, we produce a seres of results very quickly and efficiently that we can check for special characteristics. Watch.

		27  =  33
	       + 1
	        28  =  33 + 12  A perfect number
	       + 3
	        31  =  33 + 22  A prime
	       + 5
	        36  =  33 + 32  A square
	       + 7
	        43  =  33 + 42  A prime
	       + 9
	        52  =  33 + 52  Number of weeks in a year
	      + 11
	        63  =  33 + 62  ???

		etc.

     Now that's more like it, simple and orderly. So, get to work looking for your own examples of Trigg Numbers.

Problem Extension

     After reading Mr. Trigg's article (see reference below), I was attracted to the idea in a more limited way. I saw some interesting patterns in a certain group of numbers: the "Trigg squares". Here's what I saw:

100 = 62 + 43 = (6 + 4)2 = 102

225 = 102 + 53 = (10 + 5)2 = 152

     Put into words, these two examples say: The square of the N-value plus the cube of the M-value is equal to the square of the sum of N and M. And put into the symbolism of algebra, those words say:

N2 + M3 = (N + M)2

     To my delight, I found other similar instances in his results that followed that pattern. But some other squares did not "march to the same drum". Like:

1225 = 152 + 103, but (15 + 10)2 1225;

and

2304 = 242 + 123, but (24 + 12)2 2304.

     However, upon a closer examination, there is a pattern here; it is

1225 = (15 + 2×10)2 = (15 + 20)2 = 352

2304 = (24 + 2×12)2 = (24 + 24)2 = 482

     Going right to the algebra, we can write:

N2 + M3 = (N + 2 × M)2

     And there were more surprises and patterns awaiting me. So I decided to write an article (see link below) just about the squares. I won't go into all the little details at this time. Suffice it to say, you never know what treasures you have before you until you take the time to examine them closely. That's the way mathematics is for the true aficionado. That's how I feel, and I hope you do too.

Postscript

Teaching Tip:

     For those persons who can manage spreadsheets, the calculation of all the possible Trigg Numbers in this study is more easily done by that method than by the repeated addition of odd numbers explained above. I presented the addition method by way of giving a use for the "sum of consecutive odd numbers" concept as a tool for problem solving. It also has the advantange of slowing down the output of results to give the student time to reflect on what is happening. It would also be good in cooperative group learning situations; different members of the group could be assigned different "starting cubes" to work on.


References & Credits:

Charles W. Trigg, "Integers Equal to N2 + M3". Journal of Recreational Mathematics. 2(2), January 1969. pp. 44-48.

Terrel Trotter, Jr., "More About P = N2 + M3". JRM, 4(1), January 1971. pp. 45-49.

WTM is rather pleased to state that the sequence of "non-Trigg" numbers can be found in Sloane's On-line Encyclopedia of Integer Sequences and has its own reference number: A022550.

Train created by Bruce Beard.

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